Download A Game-Theoretic Model for Punctuated Equilibrium

A Game-Theoretic Model for Punctuated
Equilibrium: Species Invasion and Stasis
through Coevolution∗
Ross Cressman
Department of Mathematics
Wilfrid Laurier University
Waterloo, Ontario N2L 3C5
Canada
József Garay†
Theoretical Biology and Ecological Research Group
of Hungarian Academy of Science
and
L. Eötvös University
Department of Plant Taxonomy and Ecology
Pázmány Péter sétány1/c
H-1117 Budapest
Hungary
September, 2005
∗
This research was initiated while R.C. was a Fellow at the Collegium Budapest. He
thanks the Collegium and its staff for their hospitality and research support. The final version was finished while J.G. was a Fellow of the Konrad Lorenz Institute for Evolution and
Cognition Research. J.G. is a grantee of the Bolyai Scholarship. Financial assistance from
the Natural Sciences and Engineering Research Council of Canada, the Hungarian Science
and Technology Foundation (TET), and the Hungarian National Scientific Research Fund
(OTKA Project T037271, TO49692) is gratefully acknowledged.
†
Author of correspondence. e-mail: [email protected]
1
Abstract: A general theory of coevolution is developed that combines
the ecological effects of species’ densities with the evolutionary effects of
changing phenotypes. Our approach also treats the evolutionary changes
between coevolving species with discrete traits after the appearance of a
new species. We apply this approach to habitat selection models where new
species first emerge through competitive selection in an isolated habitat. This
successful invasion is quickly followed by evolutionary changes in behavior
when this species discovers the other habitat, leading to a punctuated equilibrium as the final outcome.
Keywords: stasis, evolutionary stability, invasion, habitat selection, discrete traits, punctuated equilibrium
2
1
Introduction
From the ecological perspective, one of the most important steps in the evolutionary process is the successful invasion of a new species into an already
existing ecosystem. Such steps are consistent with the concept of punctuated
equilibrium (Eldredge and Gould, 1972) suggested by the fossil record in paleontoecology. This theory claims that evolution is not a continuous process;
rather, during a short geological time, new species arrive in rapid succession
and contribute revolutionary morphological changes. Following these speciation events, an evolutionary stable ecosystem rapidly evolves, where lineages
are in stasis. Unfortunately, how changes occur during the speciation event
cannot be understood from the fossil record which is especially incomplete
within the short outbursts. That is, fossils tend to remain similar in appearance for many meters of vertical sediment corresponding to millions of years
of geological time. These long periods of stasis mean the morphological mutants that arose during this time could not successfully invade this existing
ecosystem and so the species look like non-changing entities.
The main purpose of this paper is to combine the essential aspects of stasis (i.e. stability) with the occasional invasion of successful mutants (either
mutant species or mutant phenotypes of existing species) in a single model.
In particular, we consider this problem of punctuated equilibrium at the
phenotypic level when phenotypic fitnesses depend on the densities of interacting species as well as on the phenotypic distribution of the system. This
contrasts with the usually assumption of evolutionary models that fitness
depends on only one factor such as the individual phenotypes, the distribution of different types in a species (Gavrilets, 2003), or the species’ densities
(i.e. population dynamics). Our coevolutionary approach also contrasts with
speciation models based on population genetics which often emphasize the
importance of reproductive isolation (Coyne, 1992; Johnson and Gullberg,
1998; Lee, 2002). Since we ignore genetic effects, our approach is closer to
ecological speciation models that have received renewed interest among both
theoretical (Schluter, 1998, 2001; Orr and Smith, 1998; Morell, 1999; McKinnon et al., 2004) and empirical (Rundle et al., 2000 Kruse et al., 2004;
Munday et al., 2004) biologists. However, our coevolutionary model also
includes changes in the phenotypic makeup of the population.
Thus, one central question of this paper is what kind of new species
and/or phenotypes are able to invade an already existing stable evolutionary
ecosystem? Here we use the term “evolutionary ecology” since, during the
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invasion, the phenotypic constitution of the currently coexisting species may
also be subject to adaptive change. There are two important aspects to this
problem. The first question is whether the new mutants can initially invade
this ecosystem when rare. When mutants are from a different species, this
first requirement guarantees that new species appear but it is not sufficient for
the long-term stability of a punctuated equilibrium since it is possible that,
after the initial successful invasion, the evolution in the enlarged system
that includes the new species selects such changes which ultimately drive
the new species to extinction. Similar theoretical instances of “evolutionary
suicide” have been studied in the literature on adaptive dynamics (Gyllenberg
and Parvinen, 2001; Diekmann, 2004). Thus, the second requirement for a
punctuated equilibrium is that the new ecosystem must be evolutionarily
stable. This corresponds to the stasis and means that further rare mutants
arising within the enlarged ecosystem must die out.
It is also clear that the successful arrival of a new species not only changes
the ecological system (e.g. a new equilibrium could arise where both resident
and invading species coexist or the resident species could be selected out),
but it may also imply an evolutionary change among the resident species that
survive. Thus, a minimal model of punctuated equilibrium must deal with
the potential of evolutionary change both within and between species.
As discussed above, a necessary step is that a new species establishes itself
into an already existing stable ecosystem. Since ecological interactions are
typically density dependent, a general theory of coevolutionary speciation
must incorporate density dependent fitness effects. An elementary example
of density dependent species’ interactions is the Lotka-Volterra model of competition between two species. For example, consider the following system,
ẋ = x(20 − 2x − y)
ẏ = y(15 − x − y).
(1)
Here x and y are the population sizes (i.e. densities) of species one and two
respectively. When only species one is present, there is an asymptotically stable equilibrium on the horizontal axis (see Figure 1). However, when species
two invades, there is an equilibrium where both species coexist at x = 5 and
y = 10. The stability properties of this equilibrium are well-known for the
Lotka-Volterra model (Goel et al., 1971; Hofbauer and Sigmund, 1998). In
particular, these properties show for system (1) that species two successfully
invades since the coexistence equilibrium is then globally asymptotically stable as in Figure 1. Here, the invasion leads to an increase in the number of
4
species in the ecosystem.
Figure 1 about here
However, in this classical Lotka-Volterra model, there is no diversity in
a given species since one of the basic assumptions of Lotka-Volterra is that
each individual in this species has the same phenotype. That is, evolutionary
change within species is impossible. Thus, this classical approach cannot
model the stasis phenomenon where rare mutants corresponding to a different
phenotype in a resident species emerge but do not successfully invade.
Our perspective to model successful invasion and stasis in a single framework is game theoretic since we believe any coevolutionary ecological scenario
must be based on individual fitness functions that depend on this individual’s
behavior (or phenotype) as well as on the behavior of other individuals in
the system (and on population density). In contrast to the Lotka-Volterra
model, this leads to changes in the phenotypic makeup of a given species and
is therefore a game theoretical situation.
A question that immediately arises is what type of evolutionary variation
in phenotypes is to be considered in the model. Maynard Smith (1960) made
a fundamental distinction between evolutionary traits (and their morphological manifestations) that are continuous (e.g. body length, weight etc.) and
that are discrete or modal (e.g. the number of fingers on a person’s hand).
The evolution of continuous traits has been widely investigated over the past
decade (see Abrams (2001) and the references therein) through models of
quantitaive genetics or phenotypic models using adaptive dynamics. Our
model of coevolutionary ecology will assume phenotypes are discrete (and
only finitely many are possible). A genetic basis for our assumption is that
the phenotype is determined by a small number of genes and so, in contrast
to continuous trait models, mutant phenotypes are not pictured as being
close to those of the resident ecosystem. On the other hand, we do not assume any knowledge of the genetic basis on which forms of mutant traits
can appear in the coevolutionary system. That is, phenotypic fitness functions will already be given in our model rather than developed from genetic
considerations. Our emphasis is then the effect these given fitnesses have on
natural selection of the population phenotypes.
Game-theoretic approaches have also been used to determine whether
within species invasion will be successful when individual fitnesses only depend on phenotypic frequencies. When applied to a resident species where
5
mutants are formed from a finite number of possible phenotypes (also called
pure strategies), the method is known as “evolutionary game theory”. In its
original form (e.g. Maynard Smith, 1982), this is a frequency dependent theory that considers a single asexual population in which there is phenotypic
diversity. The basic game-theoretic solution concept here is that of an evolutionarily stable strategy (ESS). This is a phenotype which a rare mutant
cannot invade if the overwhelming majority of the population use the ESS
(i.e. if the resident system is monomorphic and at the ESS). Since evolutionary game theory has been extended to multi-species systems (Cressman et
al., 2001; Cressman, 2005), it can also deal with the diversity and evolution
between species, but it is frequency dependent.
The game-theoretic perspective developed in this paper for a discrete set
of evolutionary traits is based on our generalization (Cressman and Garay,
2003a, 2003b) of the classical ESS conditions to N- interacting species with
frequency and density dependent fitness functions. We say an ecosystem is
evolutionarily stable if rare mutant phenotypes within these N species cannot successfully invade the ecosystem (cf. Maynard Smith’s original form).
There are two immediate consequences of mutants being “rare”. First, in
each species, only one resident and one mutant phenotype can exist at the
any given time (i.e. the rate of mutation is much slower than selection and
so after mutation the ecological system has enough time to select out the less
fit phenotypes). Second, the mutation can be produced at arbitrarily low
densities which is important since we examine invasion through a dynamical
point of view. That is, we require that mutants eventually die out under the
ecological selection dynamics. When there is only one resident species, this
leads to the intuitive requirement that each mutant phenotype is less fit than
the resident, a straightforward property that can be determined through the
fitness function. However, as we will see, when there is more than one species,
the mutants of some species may be initially more fit after they first appear
than the residents but in the long run (more precisely, in ecological time)
all mutants must die out according to the ecological dynamics. Our model
also considers invasion of an evolutionarily stable ecosystem by phenotypes
from new species. A successful invasion that leads to a different evolutionarily stable ecosystem involving the invading species then corresponds to a
punctuated equilibrium in the fossil record.
Our use of these generalized ESS stability conditions ignores the possibility an ESS may be unstable due to stochastic effects when the population
density is small (Fogel, 1997). In particular, our ecological dynamics are a
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continuous-time deterministic dynamical system. Although stochastic effects
are undoubtedly important for rare mutants to persist when they first appear, we assume the resident population is quite large (certainly, this must
be true if these phenotypes are present in the fossil record) and so unlikely
to disappear due to stochastic effects.
In Section 2, we develop a general theory of invasion and stasis. Section 3
then gives a detailed analysis of this theory applied to a two-habitat selection
model where stasis and punctuated equilibria often appear automatically.
Section 4 concludes by discussing the issues raised above in terms of the
general theory of Section 2 and the habitat selection model of Section 3.
2
Successful Invasion and Stasis
As stated in the Introduction, the fossil record is consistent with the theory
of punctuated equilibria where periods of stasis are followed by the occasional
outburst of successful invasion by mutant species and/or mutant phenotypes
of existing species. Since we want to distinguish between these two types of
invasion and since a universally agreed upon definition of "species" is still
unsettled (Wilson, 1999), we need an operative definition of species (Miller,
2001).
Here we again take the game theory perspective by placing an individual
in a given species based on its interactions with other individuals. Specifically, two individuals are in the same species if they have the same possible
behavioral phenotypes (i.e. pure strategy sets), the same interaction parameters (i.e. payoff functions) in all possible interactions, and the same basic
fitness. In mathematical game theoretical terms, they are the same “player”.
These individuals in the same species may have different ecotypes in that
they may use the possible phenotypes with different probabilities (the finite
strategy case) or they may have different measures of a quantitative phenotype such as size or body weight (the case of a continuum of strategies).
Of critical importance, individuals in the same species have the same payoff
functions. In contrast, a new species has a new payoff function based on new
phenotypes with absolutely new interactions. In game theoretical terms the
new species is a new player.
This operative definition is essential since we will ignore genetic effects
in our models by assuming asexual populations where individuals reproduce
identical offspring by some mechanism that translates payoffs directly into re7
productive success (i.e. fitness). Since no sexual reproduction is necessary in
this model, it would be possible to take each lineage to be a separate species
(corresponding to assuming every payoff function is unrelated to every other)
and so each successful mutation would lead to a new species. Although this
is an important extreme, we are more interested in modelling invasion both
within existing species as well as through the appearance of new species.
Since, from our point of view of asexual populations, individuals only differ
through their interactions, species can only be distinguished mathematically
through payoff functions based on these interactions. (This concept of species
is closely related to that defined through the generating function (G− function) approach of Vincent et al. (1993, 1996) although this literature often
requires that members of the same species be capable of interbreeding even
though genetic effects are still ignored. With a suitable bookkeeping method,
the two definitions of species are equivalent.)
However, we will also often distinguish species by the rate with which mutations occur. The more frequent type of mutation can only change phenotypes (i.e. here only the individual behavior changes and not the interaction
parameters of the payoff functions). From a biological viewpoint, this kind
of mutation introduces only new ecotypes within existing species. Although
these are the more frequent mutations, it is still assumed that the time scale
on which they occur is slower than the time scale of ecology. The most rare
mutation changes the interaction parameters as well. A new player in the
evolutionary game theory model appears in the ecosystem, corresponding to
a new biological species arising in this case. Since we suppose that both
types of mutation are very rare compared to the speed of selection in the
ecological system, at most two phenotypes can coexist in each species at a
given time. In this setting, there is a clear distinction between the selection
process (it is an ecological process), short term evolution (a process of within
species invasion by strategy mutants), and long term evolution (a process
corresponding to invasion by new players in a different species).
This assumption of separate time scales for different processes is related
to the discussions of Simpson (1944, 1955) and Mayr (1960) where the “mode
and tempo” of evolution are considered. In our terminology, the fast tempo
of the ecological process is the mode that selects the ESS of the existing
phenotypes. The second mode is through the appearance of mutations and
this occurs at a much slower tempo (actually at two tempos depending on
whether the mutants are of the same or different species). This second mode
corresponds to the emergence of novel traits.
8
Before analyzing the consequences of these assumptions concerning mutation rates, we should mention that reproductive isolation can also be used
in conjunction with the methods discussed above to distinguish between
species and is a very important mechanism for the emergence of new species
(Schluter, 2001). However, there is also the opinion (Morell, 1999; Schluter,
2001) that reproductive isolation is the result of trait (i.e. phenotype) selection. This latter ecological viewpoint fits well with the objective of this
paper, which is to study punctuated equilibria and stasis through phenotypic
evolution. In particular, we do not take into account the issue of sympatric
versus allopatric speciation that is raised by those models that emphasize
the effects of reproductive isolation.
We begin by recalling the concept of evolutionary ecological stability
(Cressman and Garay, 2003a) if there is at most one mutant in each species.
Some notation is needed for this purpose. Let there be N interacting species.
Denote by ρ = (ρ1 , ρ2 , ..., ρN ) the vector of densities of the resident phenotypes p∗ = (p1∗ , p2∗ , ..., pN∗ ) (e.g. the density of phenotype pk∗ in species k
is ρk ). Similarly, µ ≡ (µ1 , µ2 , ..., µN ) gives the densities of the mutant phenotypes p = (p1 , p2 , ..., pN ). Initially (i.e. just after mutation), the mutant
densities are all low. Together, (ρ, p∗ , µ, p) specifies the state of the ecosystem. This state is also denoted by (ρ, p∗ ) when there is only the resident
system (i.e. all µk = 0).
In this ecological system, each individual may have both intra and interspecific interactions. Thus in general, the fitness of residents, Wρk , and of
mutants, Wµk , depends on the state of the ecosystem system. Since fitness
corresponds to reproduction of identical offspring, selection is modelled by
the following Kolmogorov type dynamical system:
ρ̇k = ρk Wρk (ρ, p∗ , µ, p)
µ̇k = µk Wµk (ρ, p∗ , µ, p)
(2)
(3)
where k = 1, 2, ..., N . This system also models the situation where some
species k has only one possible phenotype (i.e. the resident phenotype) by
setting µk ≡ 0 in this case.
Remark. Dynamical systems of Kolmogorov type are often used in models of coevolution. For instance, the fitness generating functions that lead
to the static ESS Maximum Principle promoted by Vincent and co-workers
(see Cohen at al. (1999) and the references therein) are generally of this
9
type. In fact, Brown (1990) applies this principle to give a game-theoretic
interpretation to foraging behavior equilibria found by Rosenzweig (1987)
in two-habitat selection models. Our two-habitat selection model in Section 3 is also of Kolmogorov type although we assume fitness is based on
direct competition among individuals rather than indirectly through foraging on limited resources. The adaptive dynamics approach to coevolution
(e.g. Dieckmann and Law, 1996) also starts with fitness functions of Kolmogorov type before eliminating density effects and studying the “canonical
equation” of phenotypic evolution. Unlike our theory, here the density of a
monomorphic population is assumed to instantaneously track its equilibrium
value given the current phenotype (i.e. evolution occurs on the “stationary
density surface” as it is called by Cressman and Garay (2003a, 2003b)). In
contrast, our theory includes both density and phenotypic effects as follows.
We call the state (ρ∗ , p∗ ) an evolutionarily stable ecological equilibrium 1
if ρ∗ is an equilibrium of the resident system (2) with all ρ∗k > 0 and the
corresponding equilibrium (ρ∗ , p∗ , 0,p) of the system (2),(3) is locally asymptotically stable for all possible mutants p. At such an equilibrium of the
resident ecological system, no within species mutants can successfully invade
since all mutant densities evolve to 0 (i.e. mutants will all die out) according
to the above ecological dynamics. Of course, if all species have only one
possible phenotype, the ecological equilibrium is evolutionarily stable if and
only if the resident system (without mutation) is asymptotically stable at ρ∗ .
Thus, for system (1) in the Introduction, the equilibrium on the x−axis in
Figure 1 is an evolutionarily stable ecological equilibrium in this terminology
when x is equated with ρ1 , the density of the only phenotype for species one.
However, as pointed out there, this equilibrium can be successfully invaded
by species two.
In general, suppose that, in addition to the within species selection modelled by equations (2) and (3), the N−species evolutionarily stable ecological
equilibrium (ρ∗ , p∗ ) is invaded by a new player mutation whose payoff function WµN +1 is given by new interactions parameters. Let µN+1 denote its
1
In Cressman and Garay (2003a), this is called a monomorphic evolutionarily stable
ecological equilibrium since the model there also considered polymorphic cases where there
were more than one resident phenotype in some species. In Vincent et al. (1996), it is
called an ecological stable equilibrium (ESE).
10
density and pN+1 its phenotypic strategy. Then
¡
¢
ρ̇k = ρk Wρk ρ, p∗ , µ, µN+1 , p, pN+1
¡
¢
µ̇k = µk Wµk ρ, p∗ , µ, µN+1 , p, pN +1
¡
¢
µ̇N+1 = µN+1 WµN +1 ρ, p∗ , µ, µN+1 , p, pN+1
(4)
where now Wρk and Wµk for k = 1, 2, ..., N also depend on µN+1 and pN+1 .
We say that the new species with strategy pN+1 successfully invades the
resident system (ρ∗ , p∗ ) under the mutations µ if (ρ∗ , 0, 0) is an unstable
equilibrium of (4). After a successful invasion, further evolution between
species will take place since the adaptive phenotypes will change in each
species according to the new ecological situation. This may or may not lead to
stasis involving the new species. For instance, speciation does not occur if the
mutant phenotype pN +1 eventually dies out and is not replaced by any other
phenotype from species N + 1. From an evolutionary ecological viewpoint, a
punctuated equilibrium occurs if the new (N + 1)−species ecosystem has an
evolutionarily stable ecological equilibrium (i.e. an equilibrium that is stable
not only from an ecological but also from an evolutionary point of view). That
is, there must be N + 1 phenotypes (one for each of the interacting species)
p = (p1 , p2 , ..., pN , p(N+1) ) with positive densities ρ = (ρ1 , ρ2 , ..., ρN , ρN+1 )
that forms an evolutionarily stable ecological equilibrium against all possible
strategy mutants in the N + 1 species. (Stasis will also occur if this (N +
1)−species ecosystem evolves to an equilibrium that includes individuals from
species N + 1 (i.e. ρN +1 > 0) while some of the other species go extinct. We
do not discuss this possibility further here in general terms or for the habitatselection model of Section 3.)
Thus, assuming both competing species have only one possible phenotype,
Figure 1 in the Introduction illustrates a the emergence of a new species since
the successful invasion of species one (here y is equated with the density µ2
of the mutant in species two) evolves to a new stable equilibrium where both
species are present. To incorporate the stasis phenomenon into this twospecies competitive system, we must allow the possibility of other phenotypes
in the species.
Unfortunately, when there are several possible phenotypes for some species,
the general dependence of interaction parameters on the morphological phenotypic parameters is not well understood. Moreover, an adequate description of the space of all possible morphological parameters is also problematic.
Consequently, we do not discuss further directly the general theory of inva11
sion and coevolutionary speciation for the stasis and punctuated equilibria
developed by the model based on (2),(3),(4).
Instead, the following section will demonstrate our approach through the
ecological example of habitat selection. One reason for this choice is that the
habitat selection model simplifies such issues as assortative mating (which
may form the basis for the genetic process of speciation through reproductive isolation evolving in the separate habitats) (Felsenstein, 1981; Rice, 1984;
Johnson and Gullberg, 1998). Our analysis ignores this genetic process by
restricting the investigation to the evolutionary ecological aspects of the habitat selection process.
We examine situations where initially a single species is at a stable equilibrium in a two-habitat model. Here, there are two pure strategies; namely,
which of the two habitats the individual chooses. However, we are more interested in the monomorphic perspective where all resident individuals use
the same behavior strategy that specifies the proportion of their time they
spend in each habitat. In Section 3.1, we briefly summarize this model and
show when this resident system is stable against invasion by any other small
subpopulation of this species using a mutant strategy. Then, in Section 3.2,
we establish conditions when this single-species equilibrium is successfully invaded by an emerging new species. In such cases, the basic question becomes
what happens to the two-species system in the long term. Section 3.3 shows
we often arrive at a stable monomorphic system where both species coexist.
Such a phenomenon is then interpreted as a period of stasis corresponding
to a punctuated equilibrium.
3
The Two-Habitat Monomorphic Model
In Sections 3.1, 3.2 and 3.3, we determine the theoretical characterization of
evolutionarily stable ecological equilibria in general habitat selection models
with either one or two species and two habitats. Section 3.4 then develops a particular example that illustrates this theory for specific choices of
population parameters. We begin by giving common assumptions for both
models. Individuals in a given species are assumed to be morphologically indistinguishable, their only difference is how each individual divides its time
between the two habitats. Thus, the two possible pure strategies are the
choice of habitat and an individual’s phenotype is the frequencies with which
it uses the different habitats. We assume the fitness of a pure strategist in
12
a particular habitat depends only on the species’ densities in this habitat.
Finally, individuals move freely and without cost between the two habitats
and so the phenotypic frequencies (together with species’ densities in each
habitat) determine individual fitness (see, for example, (7) below).
3.1
Single Species Evolutionarily Stable Ecological Equilibrium
Let mi (i = 1, 2) be the density of this species in habitat i. Then M =
m1 + m2 is the total population size and pi = mi /M is the proportion of
the population in habitat i. The frequency vector p = (p1 , p2 ) will be called
the population (mean) strategy. It is a vector in the strategy space ∆2 ≡
{(p1 , p2 ) | p1 + p2 = 1 and pi ≥ 0}. When the population is monomorphic
(i.e. when all individuals are phenotypically identical), this p also denotes
the individual phenotype. We say the population is interior if there are
individuals in both habitats (i.e. if p is in the interior of ∆2 ).
The success Vi of an individual in habitat i is assumed to be the per
capita growth rate of a logistic equation; namely,
µ
¶
mi
Vi (m1 , m2 ) = ri 1 −
(5)
Ki
where ri is the intrinsic growth rate in habitat i and Ki is its carrying capacity. Both these parameters are positive and, when taken in combination from
the two habitats, can be interpreted as the species’ morphological parameters based on indirect competition for resources among individuals. (Notice a
change in notation from Section 2. For the general N−species theory developed there, the index k referred to the kth species. Since there will be at most
two species in Section 3, we do not index them and reserve the subscripts i, j
to indicate the habitat. Furthermore, p ∈ ∆2 denotes the phenotype of an
individual in species one rather than p1 and, in Section 3.2 where a second
species is introduced, q ∈ ∆2 is an individual phenotype in species two.)
The payoffs given by (5) will usually be written as functions of the population’s total size and mean strategy. Thus
µ
¶
pi M
Vi (p, M) = ri 1 −
.
(6)
Ki
13
Since there is costless movement between habitats, an individual with phenotype p0 has fitness
p01 V1 (p, M) + p02 V2 (p, M).
(7)
We want to check when an interior monomorphic population is “uninvadable” by any mutant monomorphic subpopulation. Let p∗ , p ∈ ∆2 denote
the behavioral strategies of the resident and mutant populations respectively
and ρ1 , µ1 ∈ R the density of resident and mutant individuals. (Notice that
we continue to denote these densities as ρ1 and µ1 to emphasize they are
variables for species 1. Although these subscripts could be dropped in this
section, this convention becomes more important when we introduce a second
species in the following section.)
Since p∗ is in the interior of ∆2 , p∗ = (p∗1 , p∗2 ) where p∗1 > 0 and p∗2 > 0.
The fitness of the resident population is
Wρ1 (ρ1 , p∗ , µ1 , p) = p∗1 V1 + p∗2 V2
µ
¶
µ
¶
ρ1 p∗1 + µ1 p1
ρ1 p∗2 + µ1 p2
∗
∗
+ p2 r2 1 −
= p1 r1 1 −
K1
K2
and the mutant fitness is
Wµ1 (ρ1 , p∗ , µ1 , p) = p1 V1 + p2 V2
µ
¶
µ
¶
ρ1 p∗1 + µ1 p1
ρ1 p∗2 + µ1 p2
+ p2 r2 1 −
= p1 r1 1 −
K1
K2
Thus the population dynamics is
³
³
´
³
´´
ρ p∗ +µ p
ρ p∗ +µ p
ρ̇1 = ρ1 p∗1 r1 1 − 1 1K1 1 1 + p∗2 r2 1 − 1 2K2 1 2
³
³
´
³
´´
ρ p∗ +µ p
ρ p∗ +µ p
µ̇1 = µ1 p1 r1 1 − 1 1K1 1 1 + p2 r2 1 − 1 2K2 1 2
(8)
If the resident population ³
cannot be´ invaded by p = (1,³0) (respectively
´
ρ p∗
ρ p∗
p = (0, 1)), we see that r1 1 − K1 11 ≤ 0 (respectively r2 1 − K1 22 ≤ 0).
³
´
ρ∗1 p∗1
∗
∗
∗
Since an equilibrium (ρ1 , 0) of (8) with ρ1 > 0 requires p1 r1 1 − K1 +
³
´
³
´ ³
´
ρ∗ p∗
ρ∗ p∗
ρ∗ p∗
p∗2 r2 1 − K1 22 = 0, we have 1 − K1 11 = 1 − K1 22 = 0. That is, an uninvadable resident population satisfies
ρ∗1 = K1 + K2
p∗i = Ki / (K1 + K2 ) .
14
(9)
This equilibrium is called the ideal free distribution (IFD) (Fretwell and
Lucas, 1970) at equilibrium density. Intuitively, each habitat is occupied to
its carrying capacity since m∗i = ρ∗1 p∗i = Ki . The equilibrium ρ∗1 is clearly
stable for the resident system
µ
µ
¶
µ
¶¶
ρ1 p∗1
ρ1 p∗2
∗
∗
+ p2 r2 1 −
ρ̇1 = ρ1 p1 r1 1 −
K1
K2
Ã
!
2
2
r1 (p∗1 )
r2 (p∗2 )
(ρ∗1 − ρ1 )
+
= ρ1
K1
K2
since ρ̇1 > 0 if and only if ρ1 < ρ∗1 . Furthermore, (ρ∗1 , 0) is asymptotically
stable for (8) although this is not so obvious since its linearization automatically has a zero eigenvalue. In fact, global asymptotic stability can be shown
from the two dimensional phase diagram for the dynamics (8) rewritten as
"Ã
!
¶ #
µ
p
p
r1 (p∗1 )2 r2 (p∗2 )2
r
r
1 1
2 2
ρ̇1 = ρ1
(ρ∗1 − ρ1 ) −
µ1
+
+
K1
K2
K1 + K2 K1 + K2
Ã
"
! #
r1 (p1 )2 r2 (p2 )2
r1 p1 + r2 p2 ∗
µ1 .
µ̇1 = µ1
(ρ1 − ρ1 ) −
+
(10)
K1 + K2
K1
K2
Specifically, the linear nullclines in the first quadrant (i.e. the lines where
ρ̇1 = 0 and µ̇1 = 0) both have negative slope and ρ̇1 = 0 is steeper than µ̇1 =
0. (A similar analysis of the nullclines for Figure 1 shows global asymptotic
stability of the coexistence equilibrium of system (1) in the Introduction.
Local asymptotic stability of (8) can also be shown by analyzing the quadratic
terms in (10) using the B-matrix approach of Cressman and Garay (2003a,
2003b). For more discussion of this technique, see also Section 3.3 below and
especially the Appendix where it is used in a four dimensional system.)
Figure 2 about here.
Thus, for the single-species habitat selection model (5), the ideal free
distribution (ρ∗1 , p∗ ) given by (9) is automatically an evolutionarily stable
ecological equilibrium.
Remark. The results of this section can also be interpreted as illustrating
stasis in a single species setting. Specifically, suppose initially all individuals
are in habitat 1. The resident system will evolve to its stasis level; namely,
15
the carrying capacity K1 in habitat 1. Introduction of a mutant phenotype
that moves between both habitats will successfully invade. Assuming the
population returns to a monomorphism, the new stasis level will be the IFD
with total carrying capacity K1 + K2 .
3.2
Invasion by a Second (Competitive) Species
In this section, we investigate the invasion of (ρ∗1 , p∗ ) by a second competitive
species. For the two-species competitive system, we generalize fitness (5) in
each habitat by taking the general Lotka-Volterra type individual fitness
functions
³
´
αi ni
i
−
Vi1 (m1 , m2 , n1 , n2 ) = ri 1 − m
i = 1, 2
Ki
Ki
³
´
(11)
β m
n
j = 1, 2.
Vj2 (m1 , m2 , n1 , n2 ) = sj 1 − Ljj − jLj j
As for the single-species model of Section 3.1, Vi1 is the individual fitness of
species one in habitat i; mi is its density; ri and Ki are the parameters of
the logistic equation (5) when the density ni of species two is 0. When ni >
0, αi is the interspecific competition coefficient for species one individuals
interacting with species two in habitat i. Analogously, Vj2 is the individual
fitness of species two in habitat j and β j is the interspecific competition
coefficient for species two individuals. Notice that the fitness parameters for
species two (i.e. sj , Lj and β j ) are in general totally unrelated to those of
species one so that species two may model a drastic morphological change
during a short evolutionary outburst following a long period of stasis. The
notation for this competitive system follows that of Cressman et al. (2004)
(see also Krivan and Sirot, 2002) for the special case where α1 = α2 = α and
β 1 = β 2 = β).
As in Section 3.1, the invasion of a monomorphism requires fitness functions be written in terms of the species’ frequencies and total population
sizes. Let M and N be the total population sizes of species one and two
respectively; vector p = (p1 , p2 ) be the mean strategy of species one (i.e.
pi is the proportion of species one in habitat i); vector q = (q1 , q2 ) be the
mean strategy of species two. (Note that we continue to use p = (p1 , p2 ) and
q = (q1 , q2 ) for the mean strategy of species one and two respectively in place
of the notation p1 and p2 in Section 2.) Then the fitness of pure strategists
16
can be rewritten as
³
Vi1 (p, M, q, N ) = ri 1 − pKi Mi −
³
q N
2
Vj (p, M, q, N ) = sj 1 − Lj j −
´
αi qi N
Ki
´
β j pj M
Lj
i = 1, 2
j = 1, 2.
(12)
An individual in species one (respectively, two) with phenotype p0 (respectively q 0 ) has fitness p01 V11 (p, M, q, N )+p02 V21 (p, M, q, N ) (respectively, q10 V12 (p, M, q, N )+
q20 V22 (p, M, q, N )).
Suppose the single-species evolutionarily stable ecological equilibrium (ρ∗1 , p∗ )
of Section 3.1 is invaded by the mutant monomorphic subpopulations of
species one and two with phenotypes p and q respectively. From (9), we
have ρ∗1 = K1 + K2 and p∗i = Ki / (K1 + K2 ). Let ρ1 , µ1 and µ2 be the
p∗ , p and q. Then, for∗ instance,
densities of individuals with phenotypes
∗
+µ1 p
∗
∗ 1 ρ1 p +µ1 p
Wρ1 (ρ1 , p , µ1 , µ2 , p, q) = p1 V1 ( ρ +µ , ρ1 + µ1 , q, µ2 ) + p∗2 V21 ( ρ1ρp +µ
, ρ1 +
1
1
1
1
µ1 , q, µ2 )). Thus, the invasion dynamics (cf. (4)) becomes
(using fitnesses Vi1
ρ1 p∗ +µ1 p
2
and Vj written in the form (12) evaluated at ( ρ +µ , ρ1 + µ1 , q, µ2 ))
1
1
¡ ∗ 1
¢
ρ̇1 = ρ1 p1 V1 + p∗2 V21
¡
¢
µ̇1 = µ1 p1 V11 + p2 V21
(13)
¡
¢
2
2
µ̇2 = µ2 q1 V1 + q2 V2 .
Since (ρ∗1 , 0) is a (globally) asymptotically stable equilibrium for the singlespecies system (i.e. when µ2 is absent or taken as µ2 ≡ 0), this invasion is
successful if the individual fitness of mutant q at this equilibrium
Wµ2 (ρ1 , p∗ , µ1 , µ2 , p, q) = q1 V12 (ρ∗1 p∗ , ρ∗1 , 0, 0) + q2 V22 (ρ∗1 p∗ , ρ∗1 , 0, 0)
is positive (and unsuccessful if negative). Here, we make the nondegeneracy assumption that the invasion fitness, Wµ2 (ρ1 , p∗ , µ1 , µ2 , p, q), of mutant
q of species two is not zero. (Note that we cannot assume this for the
species one mutant phenotype p since its fitness is automatically zero at
(ρ∗1 , 0).) Since ρ∗1 p∗i is the carrying capacity K³i of species
´ one in
³ habitat ´i,
β
K
q1 V12 (ρ∗1 p∗ , ρ∗1 , 0, 0) + q2 V22 (ρ∗1 p∗ , ρ∗1 , 0, 0) = q1 s1 1 − 1L1 1 + q2 s2 1 − β 2LK2 2 .
In summary, there will be some mutant strategy q in species two (i.e. some
q ∈ ∆2 with 0 ≤ q1 ≤ 1) that can invade the single-species evolutionarily
stable ecological equilibrium (ρ∗1 , p∗ ) if
either β 1 K1 < L1 or β 2 K2 < L2 .
17
(14)
That is, successful invasion is equivalent to the requirement that, for at least
one habitat i, the carrying capacity of species two (i.e. Li ) is larger than that
of species one adjusted by the interspecific competition coefficient of species
two (i.e. larger than β i Ki ). From the ecological viewpoint, this means that
in at least one of the habitats, one species cannot exhaust the resources from
the perspective of the other species.
3.3
Two Species Evolutionarily Stable Ecological Equilibrium and Stasis through Coevolutionary
The previous section completely characterizes through condition (14) when
there is the potential for the single species IFD of Section 3.1 to evolve to
a period of stasis involving a second species. We are particularly interested
in situations where, after successful invasion since (14) holds, the resultant
two-species system has an equilibrium where both species are present in both
habitats. Such a two-species interior equilibrium can be found by considering
the polymorphic case where each individual is a pure strategist (i.e. every
individual spends all its time in one of the habitats). Then Vi1 and Vi2 must
both be zero in (11) for i = 1, 2. The algebraic solution is
m∗i =
Ki − αi Li
Li − β i Ki
, n∗i =
, for i = 1, 2.
1 − αi β i
1 − αi β i
From (14), the positivity of both n∗1 and n∗2 implies Li > β i Ki for i = 1, 2.
That is, any mutant strategy q ∈ ∆2 of species two can successfully invade
the IFD of species one (i.e. the single-species evolutionarily stable ecological
equilibrium) given in Section 3.1. Furthermore, this positivity also implies
1 − αi β i > 0
(15)
for i = 1, 2. Inequality (15) is interesting in its own right since this is the
classical condition (cf. Hofbauer and Sigmund, 1998) that two-species coexistence is stable in each separate single-habitat model (see also the discussion
after (18) below). It also implies that Ki > αi Li (i.e. species one mutants can
successfully invade the single-species evolutionarily stable ecological equilibrium of species two). We will assume these inequalities
1 − αi β i > 0
Ki − αi Li > 0
Li − β i Ki > 0
18
(16)
for the remainder of this section unless otherwise stated.
The corresponding monomorphic equilibrium has resident strategies p∗
and q∗ in the interior of ∆2 given by
Ki − αi Li
Ki − α1 L1 + K2 − α2 L2
Li − β i Ki
=
.
Li − β 1 K1 + L2 − β 2 K2
p∗i =
q ∗i
and equilibrium densities (ρ∗1 , ρ∗2 ) = (m∗1 + m∗2 , n∗1 + n∗2 ). A two-species coevolutionary period of stasis will emerge if this equilibrium is a two-species
evolutionarily stable ecological equilibrium (i.e. it must be locally asymptotically stable for the resident system and it cannot be successfully invaded by
mutant phenotypes p and q in species one and two respectively).
Let us first get stability conditions for the resident dynamics. The resident
dynamics is
ρ̇1 = ρ1 (p∗1 V11 (ρ1 p∗1 , ρ1 p∗2 , ρ2 q ∗1 , ρ2 q∗2 ) + p∗2 V21 (ρ1 p∗1 , ρ1 p∗2 , ρ2 q ∗1 , ρ2 q ∗2 ))
ρ̇2 = ρ2 (q ∗1 V12 (ρ1 p∗1 , ρ1 p∗2 , ρ2 q ∗1 , ρ2 q∗2 ) + q∗2 V22 (ρ1 p∗1 , ρ1 p∗2 , ρ2 q∗1 , ρ2 q ∗2 ))
(17)
∗
∗
∗
(ρ¸∗1 , ρ∗2 ) ·= (m∗1 + m
with
equilibrium
2 , n1 + n2 ). The linearized dynamics is
·
¸
· ∗
¸
ρ̇1 ∼ ρ1 0
ρ1 − ρ∗1
J RR
where J RR has entries
=
∗
0 ρ2
ρ̇2
ρ2 − ρ∗2
JijRR ≡
∂ (ρ̇i /ρi )
.
∂ρj
The calculation of these partials evaluated at the equilibrium yields
¸
·
rb1 p∗2
b2 p∗2
α1 rb1 p∗1 q ∗1 + α2 rb2 p∗2 q∗2
RR
1 +r
2
−J =
β 1 sb1 p∗1 q ∗1 + β 2 sb2 p∗2 q∗2
sb1 q∗2
b2 q∗2
1 +s
2
with rbi ≡ Krii and sbi ≡ Lsii . Assuming nondegeneracy, since the diagonal
entries of J RR are negative, the resident system is asymptotically stable if
and only if
¡
¢ ¡ ∗2
¢
det J RR = rb1 p∗2
b2 p∗2
sb1 q 1 + sb2 q ∗2
1 +r
2
2
− (α1 rb1 p∗1 q ∗1 + α2 rb2 p∗2 q ∗2 ) (β 1 sb1 p∗1 q ∗1 + β 2 sb2 p∗2 q∗2 )
∗2
∗2
b2 sb2 p∗2
= rb1 sb1 p∗2
(18)
1 q 1 (1 − α1 β 1 ) + r
2 q 2 (1 − α2 β 2 )
∗2 ∗2
∗ ∗ ∗ ∗
∗2 ∗2
∗ ∗ ∗ ∗
+b
r1 sb2 (p1 q 2 − α1 β 2 p1 q 1 p2 q 2 ) + rb2 sb1 (p2 q1 − α2 β 1 p1 q 1 p2 q2 )
> 0.
19
From (18), the stability of the resident system is not the same as stability of the two separate habitat models (which, from (15) has the exact
requirement 1 − αi β i > 0 for i = 1, 2). In particular, there are two-habitat
systems where a single-species evolutionarily stable ecological equilibrium
can be successfully invaded by a second species and the resultant two-species
two-habitat system has an equilibrium with both species present in habitats
1 and 2 but this equilibrium does not yield a period of stasis since it is not
a two-species evolutionarily stable ecological equilibrium.
For the remainder of this section, suppose the resident two-species system
is stable at an interior equilibrium (i.e. all the inequalities in (16) and (18)
hold). For instance, this will happen if the products αi β j of interspecific
competition coefficients are small for all i and j. For this to be a twospecies evolutionarily stable ecological equilibrium, we require it to be locally
asymptotically stable under invasion by any mutant strategies p and q in
species one and two respectively. From (2) and (3) using the notation of
Section 3, we have the four-dimensional system
¶
µ ∗ 1
p1 V1 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q∗1 + µ2 q1 , ρ2 q ∗2 + µ2 q2 )
ρ̇1 = ρ1
+p∗ V 1 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q∗1 + µ2 q1 , ρ2 q ∗2 + µ2 q2 )
¶
µ ∗ 2 22
q1 V1 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q ∗1 + µ2 q1 , ρ2 q∗2 + µ2 q2 )
ρ̇2 = ρ2
(19)
+q ∗2 V22 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q ∗1 + µ2 q1 , ρ2 q∗2 + µ2 q2 )
¶
µ
p1 V11 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q ∗1 + µ2 q1 , ρ2 q∗2 + µ2 q2 )
µ̇1 = µ1
+p2 V21 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q ∗1 + µ2 q1 , ρ2 q∗2 + µ2 q2 )
µ
¶
q1 V12 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q ∗1 + µ2 q1 , ρ2 q ∗2 + µ2 q2 )
µ̇2 = µ2
.
+q2 V22 (ρ1 p∗1 + µ1 p1 , ρ1 p∗2 + µ1 p2 , ρ2 q ∗1 + µ2 q1 , ρ2 q ∗2 + µ2 q2 )
From the Appendix, we find that two-species stability of the monomorphic
resident system (that has mutual coexistence in both habitats) implies the
resident system is uninvadable by any monomorphic subsystem (p, q).2 That
is, inequalities (16) and (18) characterize the existence of an interior twospecies evolutionarily stable ecological equilibrium. For the habitat selection
model, Cressman et al. (2004) call this equilibrium a two-species ideal free
2
Notice that this is the same result as for a single species (in Section 3.1) except for
single species the monomorphic resident system with both habitats occupied is globally
asymptotically stable through a phase plane analysis of the two-dimensional dynamical
system. For two-species, the four-dimensional phase portrait is intractable and only local
asymptotic stability is shown in the Appendix by using B-matrices.
20
distribution at equilibrium densities, in analogy to the single-species IFD of
Fretwell and Lucas (1970).
In summary, the single-species IFD of Section 3.1 becomes a punctuated
equilibrium in the long run for the two-habitat selection model whenever an
interior two-species IFD exists.
3.4
An Example
This section examines evolutionarily stable ecological equilibria for a particular set of population parameters in (12) in order to illustrate the general theory developed in Sections 3.1, 3.2 and 3.3 for two-habitat selection.
Specifically, for species one, let ri = 20, Ki = 5 and αi = 12 for i = 1, 2. In
particular, the two habitats are identical as far as species one is concerned.
They are also identical for species two where we take si = 15, Li = 15
and
2
β i = 1 for i = 1, 2. This set of parameters was chosen in such a way that
the existence of a two-species evolutionarily stable ecological equilibrium is
closely related to the Lotka-Volterra competition system (1) in the Introduction (see system (20) below).
Following Section 3.1, the (single-species)
ideal free distribution (9) is
¡
¢
(ρ∗1 , p∗ ) where ρ∗1 = 10 and p∗ = 12 , 12 . This evolutionarily stable ecological
equilibrium is a monomorphism that successfully invades the unstable equilibrium where all individuals are in only one of the habitats and density is
5. See Figure 2. The latter unstable equilibrium corresponds to a period
of stasis before this species discovers the other habitat and the population
evolves quickly to its carrying capacity in both habitats.
From Section 3.2, the second species will successfully invade (ρ∗1 , p∗ ) since
= Li ). In fact, since both incondition (14) is satisfied (i.e. β i Ki = 5 < 15
2
equalities in (14) hold, any monomorphic species two population with strategy q = (q1 , q2 ) (i.e. each individual of species two spends a proportion qi of
its time in habitat i) can invade. However, it seems more plausible that the
initial invaders from species two will only appear in one of the habitats, say
habitat 1 and so q = (1, 0). With our set of parameters, the new monomorphic equilibrium from (12) with species one in both habitats
¡ 1 and
¢ species two
15
2
only in habitat 1 is then (ρ1 , p, ρ2 , q) with ρ1 = 2 , p = 3 , 3 , ρ2 = 5 and
q = (1, 0). As a resident system, this equilibrium is stable and so represents
another period of stasis until individuals in species two discover the benefit
of spending part of their time in habitat 2. That is, (ρ∗1 , p∗ ) has been replaced
temporarily by (ρ1 , p, ρ2 , q). In other words, the single-species evolutionarily
21
¡1 1¢
∗
stable ecological equilibrium strategy
¡ 1 2 ¢ p = 2 , 2 is completely eliminated
by the monomorphism at p = 3 , 3 through the introduction of species two
individuals in habitat 1.
Figure 3 about here.
The above application of the theory from Section 3.2 does not mean the
new (two-species) equilibrium (ρ1 , p, ρ2 , q) is an evolutionarily stable ecological equilibrium. However, from (12), there is in fact an interior equilibrium
where both species
are present in both
habitats;
namely, (ρ∗1 , p∗ , ρ∗2 , q ∗ ) where
¡
¡
¢
¢
ρ∗1 = 5, p∗ = 12 , 12 , ρ∗2 = 10, q∗ = 12 , 12 . From (17), the dynamical system
for the resident monomorphic populations with phenotypes p∗ and q∗ is
ρ̇1 = ρ1 (20 − 2ρ1 − ρ2 )
ρ̇2 = ρ2 (15 − ρ1 − ρ2 ) .
(20)
This system is identical to (1) and so, from the Introduction, the resident
system is globally asymptotically stable (see Figure 1 there). From Section
3.3, it is also asymptotically stable with respect to simultaneous invasion
by monomorphic subpopulations in both species. Thus, (ρ∗1 , p∗ , ρ∗2 , q∗ ) is an
evolutionarily stable ecological equilibrium corresponding to a punctuated
equilibrium involving two species. Stasis will be maintained at least until
this equilibrium is successfully invaded by a rare mutant from a third species.
4
Conclusion
This paper develops a general theory of invasion and stasis through coevolution by combining the ecological effects of species’ densities with the evolutionary effects of changing phenotypes. One step in this process is the
successful invasion by a new monomorphic species with a fixed phenotype.
Since phenotypes are fixed and only densities vary in this initial step, the successful invasion is then an instance of ecological speciation as described by
Schluter (2001). However, this ecological speciation approach does not take
account of the evolutionary effects the new species will eventually have on
the phenotypic makeup of the system. That is, ecological speciation is only
concerned with short term stasis. Our coevolutionary theory goes further by
modelling the evolutionary effects through the game-theoretical method of
individual fitness functions depending on both the density and phenotypic
22
makeup of the species. Here, long term stasis corresponds to resident systems
where no monomorphic subpopulations exhibiting fixed mutant phenotypes
can successfully invade. Long term stasis is then given by an evolutionarily
stable ecological equilibrium which persists until successfully invaded by another new species. If successful new species rarely appear, this process can
produce a sequence of punctuated equilibria where there are long periods of
stasis separated by short outbursts of evolving ecosystems.
Our theory of punctuated equilibria does not include population genetics
explicitly as it is based on the standard assumption of evolutionary game
theory that individual fitness translates into reproduction of offspring with
identical phenotypes. On the other hand, one of the cornerstones of many
speciation models is the reproductive isolation of subpopulations that have
adapted genetically either through assortative mating (sympatric speciation)
or through physical separation (allopatric speciation). Clearly, population
genetics plays an important role in speciation as an intuitive requirement
that two species are different is that they cannot interbreed.
However, it is not so clear whether this reproductive separation is the
initial cause of speciation or simply the end result of ecological or evolutionary
factors. Take, for example, the two-habitat selection model of Section 3.
Assuming mating occurs only between individuals in the same habitat, the
indirect competition between individuals to settle in the better habitat results
in assortative mating without involving population genetics. It seems here
that ecological factors favoring the emergence of new species are paramount
and reproductive isolation is only a byproduct.
This habitat selection model also illustrates nicely the process of invasion
by species two into one of the habitats followed by the phenotypic evolution of the resultant two species system to an evolutionarily stable ecological
equilibrium (i.e. the ensuing coevolutionary effects) where both habitats are
occupied by each species. The methods developed here can be extended to
coevolutionary speciation events where successful new species drive (some)
existing species to extinction. The fossil record seems to indicate successful
invasion is often combined with mass extinctions. Outside these drastic perturbations, it is an open question in biology whether ecological or genetical
factors are more important in keeping the number of species relatively constant (Charles et al., 2001). Our analysis supports the relevance of ecology
in such matters.
23
5
Appendix
For the single-species two-habitat model of Section 3.1, the invasion of the
evolutionarily stable ecological equilibrium (ρ∗1 , p∗ ) by a monomorphic subpopulation, where all individuals are using strategy p, leads to the dynamical
system (10) whose linearization about (ρ1 , µ1 ) = (ρ∗1 , 0) automatically has a
zero eigenvalue since ∂∂µµ̇1 = 0. In this situation, Cressman and Garay (2003a,
1
2003b) say the invasion is by a selectively neutral mutant strategy. Similarly,
the two-species evolutionarily stable ecological equilibrium (ρ∗1 , p∗ , ρ∗2 , q ∗ ) for
the two-habitat model of Section 3.3 is simultaneously invaded by selectively
neutral mutant strategies by both species (i.e. all of the corresponding eigenvalues are zero). The dynamical systems, (10) and (19) respectively, for both
the single-species and two-species invasion of the resident systems are of the
form
ρ̇i = ρi fi (ρ, µ)
(21)
µ̇i = µi gi (ρ, µ)
where i = 1 for a single species and i = 1, 2 for a two species system.
Although asymptotic stability of (ρ∗1 , p∗ ) can be shown by a phase plane
analysis for a single species (see Figure 2), this is no longer possible for the
two-species four-dimensional dynamical system (19). Another method that
is applicable to both systems is the expansion of (21) about the equilibrium
up to quadratic terms. Following Cressman and Garay (2003a, 2003b), the
four Jacobian matrices
JijRR ≡
JijIR ≡
∂(ρ̇i /ρi )
∂ρj
∂(µ̇i /µi )
∂ρj
JijRI ≡
JijII ≡
∂(ρ̇i /ρi )
∂µj
∂(µ̇i /µi )
∂µj
are used where these partial derivatives are evaluated at the equilibrium.
They showed that the equilibrium is asymptotically stable if and only if the
resident system is asymptotically stable (i.e. the eigenvalues of J RR have
¡
¢−1 RI
J is a B-matrix.
negative real parts) and J ≡ J II − J IR J RR
Remark. The B-matrix conditions for an N × N matrix J are given
in Hofbauer and Sigmund (1998). For N = 1, the only entry of J must be
negative. For N = 2, we need both diagonal entries of J being negative
and its determinant positive. If the N × N matrix J satisfies the degenerate
condition that it has a zero
(i.e. a nonzero vector X with nonnegative
Pray
N
components such that Xi j=1 Jij Xj = 0 for all i), then the equilibrium
24
may be asymptotically stable without J being a B-matrix (Cressman and
Garay, 2003b). For the general speciation and invasion model of Section 2,
the mutants need not be selectively neutral. Any mutant strategy that is not
selectively neutral must be at a selective disadvantage (i.e. its corresponding
eigenvalue must be negative) or else it can successfully invade the resident
system.
For the single-species two-habitat model of Section 3.1, each Jacobian
matrix has a single entry given by
¶
µ
´
³ ∗
2
2
r1 (p∗1 )
r2 (p∗2 )
r1 p1 p1
r2 p∗2 p2
RR
RI
≡
−
J ≡−
J
+
+
K1
K2
K1
K2
´
´ .
³ ∗
³
2
∗
r1 p1 p1
r2 p2 p2
r1 (p1 )
r2 (p2 )2
IR
II
J ≡−
+ K2
J ≡ − K1 + K2
K1
The eigenvalue of J
RR
, −
µ
2
r1 (p∗1 )
K1
2
+
r2 (p∗2 )
K2
¶
, is clearly negative. Thus,
(ρ∗1 , 0) is asymptotically stable if and only if the only entry of J is negative. Here
Ã
! ³ ³ r1 p∗1 p1 r2 p∗2 p2 ´´2
2
2
− K1 + K2
r1 (p1 )
r2 (p2 )
¶
− µ ∗ 2
+
J = −
2
K1
K2
r1 (p1 )
r2 (p∗2 )
−
+ K2
K1
= −
r1 r2
(p∗ p2 − p∗2 p1 )2 .
K1 K2 1
This is negative if and only if p 6= p∗ . That is, (ρ∗1 , p∗ ) is uninvadable by any
other strategy p in this single species.
To analyze the asymptotic stability of the equilibrium (ρ∗1 , ρ∗2 , 0, 0) of the
two-species system (19), we assume (ρ∗1 , ρ∗2 ) is an asymptotically stable equilibrium of the resident system (17) and find the 2 × 2 Jacobian matrices.
These are given by (note that the matrix J RR is already given in Section 3.3)
¸
·
rb1 p∗2
b2 p∗2
α1 rb1 p∗1 q ∗1 + α2 rb2 p∗2 q∗2
RR
1 +r
2
−J =
β 1 sb1 p∗1 q ∗1 + β 2 sb2 p∗2 q∗2
sb1 q∗2
b2 q∗2
1 +s
2
¸
·
rb1 p∗1 p1 + rb2 p∗2 p2
α1 rb1 p∗1 q1 + α2 rb2 p∗2 q2
RI
−J =
β 1 sb1 p1 q∗1 + β 2 sb2 p2 q ∗2
sb1 q ∗1 q1 + sb2 q ∗2 q2
¸
·
rb1 p1 p∗1 + rb2 p2 p∗2
α1 rb1 p1 q ∗1 + α2 rb2 p2 q ∗2
IR
−J =
β 1 sb1 p∗1 q1 + β 2 sb2 p∗2 q2
sb1 q1 q ∗1 + sb2 q2 q ∗2
25
−J
II
=
·
rb1 p21 + rb2 p22
α1 rb1 p1 q1 + α2 rb2 p2 q2
β 1 sb1 p1 q1 + β 2 sb2 p2 q2
sb1 q12 + sb2 q22
¸
.
We start with the diagonal terms of −J. With
·
¸
¡ RR ¢−1
1
−b
α1 rb1 p∗1 q∗1 + α2 rb2 p∗2 q ∗2
b2 q∗2
s1 q ∗2
1 −s
2
,
=
J
−b
r1 p∗2
b2 p∗2
det J RR β 1 sb1 p∗1 q ∗1 + β 2 sb2 p∗2 q ∗2
1 −r
2
we find (−J)11 is equal to rb1 p21 + rb2 p22 plus the matrix product
·
¸
£
¤ ¡ RR ¢−1
rb1 p∗1 p1 + rb2 p∗2 p2
∗
∗
∗
∗
rb1 p1 p1 + rb2 p2 p2 α1 rb1 p1 q 1 + α2 rb2 p2 q 2 J
.
β 1 sb1 p1 q ∗1 + β 2 sb2 p2 q ∗2
¢
¡
Thus det J RR (−J)11 is equal to
¸
·
∗2
∗2
¢
¡ 2
rb1 sb1 p∗2
b2 sb2 p∗2
2
1 q 1 (1 − α1 β 1 ) + r
2 q 2 (1 − α2 β 2 )
rb1 p1 + rb2 p2
∗2
∗ ∗ ∗ ∗
∗2
∗ ∗ ∗ ∗
+b
r1 sb2 (p∗2
b2 sb1 (p∗2
1 q 2 − α1 β 2 p1 q 1 p2 q 2 ) + r
2 q 1 − α2 β 1 p1 q 1 p2 q 2 )
plus the matrix product
£
rb1 p1 p∗1 + rb2 p2 p∗2 α1 rb1 p∗1 q∗1 + α2 rb2 p2 q ∗2
 ·
¤


− (b
s1 q ∗2
b2 q∗2
r1 p∗1 p1 + rb2 p∗2 p2 )
1 +s
2 ) (b
∗ ∗
∗ ∗
+·(α1 rb1 p1 q1 + α2 rb2 p2 q 2 ) (β 1 sb1 p1 q ∗1 + β 2 sb2 p2 q¸∗2 )
(β 1 sb1 p∗1 q ∗1 + β 2 sb2 p∗2 q ∗2 ) (b
r1 p∗1 p1 + rb2 p∗2 p2 )
∗2
∗2
− (b
r1 p1 + rb2 p2 ) (β 1 sb1 p1 q∗1 + β 2 sb2 p2 q ∗2 )
¡
¢
∗ 2
s1 q ∗2
b2 q∗2
This simplifies to det J RR (−J)11 = rb1 rb2 [b
1 (1 − α1 β 1 ) + s
2 (1 − α2 β 2 )] (p1 − p1 ) .
¡
¢
∗ 2
r1 p∗2
b2 p∗2
Similarly det J RR (−J)22 = sb1 sb2 [b
1 (1 − α1 β 1 ) + r
2 (1 − α2 β 2 )] (q1 − q 1 ) .
Since 1 − αi β i < 0 by (16), the diagonal terms are negative as required for J
to be a B-matrix.
To calculate the determinant of J, we also need the off diagonal terms.
Adapting
¡
¢ the above steps, we find
RR
∗ ∗
− α1 β 1 ) + α1 sb2 p∗2 q ∗2 (1 − α2 β 2 )] (p1 − p∗1 ) (q1 − q ∗1 )
det¡J
(−J)
¢ 12 = rb1 rb2 [α2 sb1 p1 q 1 (1
∗ ∗
RR
β 1 rb2 p∗2 q ∗2 (1 − α2 β 2 )] (p1 − p∗1 ) (q1 − q ∗1 ).
(−J)21 = sb1 sb2 [β
and det J
1β 1) + ¸
· 2 rb1 p1 q1 (1 − α
2
2
rb1 rb2 sb1 sb2 (p1 −p∗1 ) (q1 −q ∗1 )
Thus det J is the product of
and
(det J RR )2
·
(b
s1 q ∗2
b2 q ∗2
r1 p∗2
b2 p∗2
1 (1 − α1 β 1 ) + s
2 (1 − α2 β 2 )) (b
1 (1 − α1 β 1 ) + r
2 (1 − α2 β 2 ))
∗ ∗
∗ ∗
∗ ∗
− (α2 sb1 p1 q1 (1 − α1 β 1 ) + α1 sb2 p2 q2 (1 − α2 β 2 )) (β 2 rb1 p1 q 1 (1 − α1 β 1 ) + β 1 rb2 p∗2 q∗2 (1 − α2 β 2 ))
Since
b2 q ∗2
r1 p∗2
b2 p∗2
(b
s1 q∗2
1 (1 − α1 β 1 ) + s
2 (1 − α2 β 2 )) (b
1 (1 − α1 β 1 ) + r
2 (1 − α2 β 2 ))
∗ ∗
∗ ∗
∗ ∗
− (α2 sb1 p1 q 1 (1 − α1 β 1 ) + α1 sb2 p2 q 2 (1 − α2 β 2 )) (β 2 rb1 p1 q1 (1 − α1 β 1 ) + β 1 rb2 p∗2 q∗2 (1 − α2 β 2 ))
26
¸ 

.

¸
.
2
(1−α1 β )(1−α2 β )b
r1 rb2 sb1 sb2 (p1 −p∗1 )
1
2
is equal to (1 − α1 β 1 ) (1 − α2 β 2 ) det J RR , we have det J =
det J RR
In particular, det J > 0 if and only if det J RR > 0. That is, (ρ∗1 , ρ∗2 , 0, 0) is
asymptotically stable for (19) if and only if (ρ∗1 , ρ∗2 ) is asymptotically stable
¥
for (17).
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30
20
y 10
0
10
x
20
Figure 1. The phase diagram for the Lotka-Volterra system (1). Four
trajectories are plotted (the solid curves evolving to the interior equilibrium
(5, 10)) as well as the isoclines through this equilibrium (the two dotted lines).
31
20
mu1 10
0
10
rho1
20
Figure 2. The phase diagram for the invasion dynamics (10) of the singlespecies IFD (the system parameters used are those of the example in Section
3.4). Four trajectories are plotted (the solid curves evolving to the boundary
equilibrium (10, 0) corresponding to the IFD (ρ∗1 , p∗ ) = (10, ( 12 , 12 ))) as well as
the isoclines through this equilibrium (the two dotted lines). The top isocline
is where ρ̇1 = 0 and the bottom where µ̇1 = 0.
32
10
mu2
0
mu1
rho1
10
10
Figure 3. The phase diagram for system (13) using the system parameters of Section 3.4 when the monomorphic single-species IFD at (ρ∗1 , p∗ ) =
(10, ( 12 , 12 )) is invaded by the mutant monomorphic subpopulations p = ( 13 , 23 )
(corresponding to density µ1 ) and q = (1, 0) (corresponding to density µ2 ).
One trajectory is shown that starts near this IFD (i.e. near (ρ1 , µ1 , µ2 ) =
(10, 0, 0)), evolving to the two-species equilibrium at (0, 15
, 5) where p∗ has
2
been completely eliminated. In fact, this two-species equilibrium is globally asymptotically stable for the system with individual strategies p∗ , p, q as
indicated by the other three trajectories evolving to it from various initial
points in the phase diagram. This equilibrium results from an invasion by
species two in habitat one and will eventually be replaced by a two-species
evolutionarily stable equilibrium where both species coexist in each habitat.
33